Suppose given an extension of profinite groups,
1 \to G \to \Gamma \xrightarrow {\deg } \widehat{\mathbf{Z}} \to 1
We say \Gamma has an abstract trace formula if and only if there exist
an integer q\geq 1, and
for every d\geq 1 a finite set S_ d and for each x\in S_ d a conjugacy class F_ x \in \Gamma with \deg (F_ x) = d
such that the following hold
for all \ell not dividing q have \text{cd}_\ell (G)<\infty , and
for all finite rings \Lambda with q\in \Lambda ^*, for all finite projective \Lambda -modules M with continuous \Gamma -action, for all n > 0 we have
\sum \nolimits _{d|n}d \left( \sum \nolimits _{x \in S_ d} \text{Tr}( F_ x^{n/d} |_ M) \right) = q^ n \text{Tr}(F^ n|_{M \otimes _{\Lambda [[G]]}^{\mathbf{L}}\Lambda })
in \Lambda ^\natural .
Here M \otimes _{\Lambda [[G]]}^{\mathbf{L}}\Lambda = LH_0(G, M) denotes derived homology, and F=1 in \Gamma /G = \widehat{\mathbf{Z}}.
Example 64.30.2. Fix an integer q\geq 1
\begin{matrix} 1
& \to
& G = \widehat{\mathbf{Z}}^{(q)}
& \to
& \Gamma
& \to
& \widehat{\mathbf{Z}}
& \to
& 1
\\ & & = \prod _{l\not\mid q} \mathbf{Z}_ l
& & F
& \mapsto
& 1
\end{matrix}
with FxF^{-1} = ux, u \in (\widehat{\mathbf{Z}}^{(q)})^*. Just using the trivial modules \mathbf{Z}/m\mathbf{Z} we see
q^ n - (qu)^ n \equiv \sum \nolimits _{d|n} d\# S_ d
in \mathbf{Z}/m\mathbf{Z} for all (m, q)=1 (up to u \to u^{-1}) this implies qu = a\in \mathbf{Z} and |a| < q. The special case a = 1 does occur with
\Gamma = \pi _1^ t(\mathbf{G}_{m, \mathbf{F}_ p}, \overline\eta ), \quad \# S_1 = q - 1, \quad \text{and}\quad \# S_2 = \frac{(q^2-1)-(q-1)}{2}
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